Find sin theta if cot theta =-4 and cos theta < 0

Imagine the unit circle. The cot(theta) is a line from (0,1) to (-4,1). Imagine it is part of a triangle with the origin (draw it!).

Then the hypotenuse length is √(1+4²) = √17.

The sine rule says that sin(90)/√17 must equal sin(theta)/4, and sin(90)=1, so